Square Roots and Fractions

Date: 10/27/98 at 08:49:33
From: Doctor Peterson
Subject: Re: Dividing Equations
Okay, Billy, presumably what you are supposed to do is to simplify the
expression. There's nothing you can do to eliminate the square root
(other than actually evaluating it, which would be only an
approximation), but we can simplify it so that we only take the square
root of an integer (not a fraction), and all square roots are in the
numerator (so that when we evaluate it we won't have to divide by a
lot of digits). Simplifying radical expressions like this sometimes
also requires making the number inside the square root sign as small
as possible, but we won't have to do that in this problem.
In e-mail, we write the square root as "sqrt", so your problem becomes:
Simplify sqrt(1/2)
What we do is to first use the rule that:
sqrt(a/b) = sqrt(a) / sqrt(b)
so we have:
sqrt(1) 1
sqrt(1/2) = ------- = -------
sqrt(2) sqrt(2)
Now we only have the square root of an integer, but it's in the
denominator. How can we move it? If we multiply sqrt(2) by itself, we
get an integer, 2; so let's multiply both numerator and denominator by
sqrt(2), which won't change the value:
1 1 * sqrt(2) sqrt(2)
------- = ----------------- = -------
sqrt(2) sqrt(2) * sqrt(2) 2
That's the answer. If you want to go on and evaluate it, you don't even
need a calculator, if you know that approximately sqrt(2) = 1.414. This
tells us that sqrt(1/2) is about 0.707. That would have been much
harder to figure out without simplifying first.
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/